Explicit Error Bounds for Carleman Linearization
For researchers using Carleman linearization, this fills a gap by providing explicit, computable error bounds, though the approach is incremental.
The paper provides explicit error bounds for Carleman linearization of polynomial ODEs, proposing two strategies: one via iterative backwards-integration requiring an a priori solution estimate, and another combinatorial approach using generating functions that yields a computable local error bound.
We revisit the method of Carleman linearization for systems of ordinary differential equations with polynomial right-hand sides. This transformation provides an approximate linearization in a higher-dimensional space through the exact embedding of polynomial nonlinearities into an infinite-dimensional linear system, which is then truncated to obtain a finite-dimensional representation with an additive error. To the best of our knowledge, no explicit calculation of the error bound has been studied. In this paper, we propose two strategies to obtain a time-dependent function that locally bounds the truncation error. In the first approach, we proceed by iterative backwards-integration of the truncated system. However, the resulting error bound requires an a priori estimate of the norm of the exact solution for the given time horizon. To overcome this difficulty, we construct a combinatorial approach and solve it using generating functions, obtaining a local error bound that can be computed effectively.